where the (unknown) function x=x(t){\displaystyle x=x(t)} is the displacement at time t,{\displaystyle t,}x˙{\displaystyle {\dot {x}}} is the first derivative of x{\displaystyle x} with respect to time, i.e. velocity, and x¨{\displaystyle {\ddot {x}}} is the second time-derivative of x,{\displaystyle x,} i.e. acceleration. The numbers δ,{\displaystyle \delta ,}α,{\displaystyle \alpha ,}β,{\displaystyle \beta ,}γ{\displaystyle \gamma } and ω{\displaystyle \omega } are given constants.

The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.

The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The restoring force provided by the nonlinear spring is then αx+βx3.{\displaystyle \alpha x+\beta x^{3}.}

When α>0{\displaystyle \alpha >0} and β>0{\displaystyle \beta >0} the spring is called a hardening spring. Conversely, for β<0{\displaystyle \beta <0} it is a softening spring (still with α>0{\displaystyle \alpha >0}). Consequently, the adjectives hardening and softening are used with respect to the Duffing equation in general, dependent on the values of β{\displaystyle \beta } (and α{\displaystyle \alpha }).[1]

The number of parameters in the Duffing equation can be reduced by two through scaling, e.g. the excursion x{\displaystyle x} and time t{\displaystyle t} can be scaled as:[2]τ=tα{\displaystyle \tau =t{\sqrt {\alpha }}} and y=xα/γ,{\displaystyle y=x\alpha /\gamma ,} assuming α{\displaystyle \alpha } is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then:[3]

The dots denote differentiation of y(τ){\displaystyle y(\tau )} with respect to τ.{\displaystyle \tau .} This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters (ε,{\displaystyle \varepsilon ,}η{\displaystyle \eta } and σ{\displaystyle \sigma }) and two initial conditions (i.e. for y(t0){\displaystyle y(t_{0})} and y˙(t0){\displaystyle {\dot {y}}(t_{0})}).

since δ≥0{\displaystyle \delta \geq 0} for damping. Without forcing the damped Duffing oscillator will end up at (one of) its stableequilibrium point(s). The equilibrium points, stable and unstable, are at αx+βx3=0.{\displaystyle \alpha x+\beta x^{3}=0.} If α>0{\displaystyle \alpha >0} the stable equilibrium is at x=0.{\displaystyle x=0.} If α<0{\displaystyle \alpha <0} and β>0{\displaystyle \beta >0} the stable equilibria are at x=+−α/β{\displaystyle x=+{\sqrt {-\alpha /\beta }}} and x=−−α/β.{\displaystyle x=-{\sqrt {-\alpha /\beta }}.}

The frequency response of this oscillator describes the amplitudez{\displaystyle z} of steady state response of the equation (i.e. x(t){\displaystyle x(t)}) at a given frequency of excitation ω.{\displaystyle \omega .} For a linear oscillator with β=0,{\displaystyle \beta =0,} the frequency response is also linear. However, for a nonzero cubic coefficient, the frequency response becomes nonlinear. Depending on the type of nonlinearity, the Duffing oscillator can show hardening, softening or mixed hardening–softening frequency response. Anyway, using the homotopy analysis method or harmonic balance, one can derive a frequency response equation in the following form:[9][5]

For certain ranges of the parameters in the Duffing equation, the frequency response may no longer be a single-valued function of forcing frequency ω.{\displaystyle \omega .} For a hardening spring oscillator (α>0{\displaystyle \alpha >0} and large enough positive β>βc+>0{\displaystyle \beta >\beta _{c+}>0}) the frequency response overhangs to the high-frequency side, and to the low-frequency side for the softening spring oscillator (α>0{\displaystyle \alpha >0} and β<βc−<0{\displaystyle \beta <\beta _{c-}<0}). The lower overhanging side is unstable – i.e. the dashed-line parts in the figures of the frequency response – and cannot be realized for a sustained time. Consequently, the jump phenomenon shows up:

when the angular frequency ω{\displaystyle \omega } is slowly increased (with other parameters fixed), the response amplitudez{\displaystyle z} drops at A suddenly to B,

if the frequency ω{\displaystyle \omega } is slowly decreased, then at C the amplitude jumps up to D, thereafter following the upper branch of the frequency response.

The jumps A–B and C–D do not coincide, so the system shows hysteresis depending on the frequency sweep direction.[9]

Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below. The forcing amplitude increases from γ=0.20{\displaystyle \gamma =0.20} to γ=0.65.{\displaystyle \gamma =0.65.} The other parameters have the values: α=−1,{\displaystyle \alpha =-1,}β=+1,{\displaystyle \beta =+1,}δ=0.3{\displaystyle \delta =0.3} and ω=1.2.{\displaystyle \omega =1.2.} The initial conditions are x(0)=1{\displaystyle x(0)=1} and x˙(0)=0.{\displaystyle {\dot {x}}(0)=0.} The red dots in the phase portraits are at times t{\displaystyle t} which are an integer multiple of the periodT=2π/ω.{\displaystyle T=2\pi /\omega .}[10]